\[\text{PV} = \frac{c}{1+r} + \frac{c}{(1+r)^2} + \frac{c}{(1+r)^3} + \cdots\]

• The value is \(c/r\).
• Example: the value of getting $100 forever, starting in one year, when the rate of return is 10%, is $1,000.
• Proof 1: sum of geometric series
• Proof 2: if you invest $1,000 and earn 10%, you can withdraw $100 in a year without depleting your capital, and continue doing so forever.

\[\text{PV} = \frac{c}{1+r} + \frac{(1+g)c}{(1+r)^2} + \frac{(1+g)^2c}{(1+r)^3} + \cdots \]

• The value is \(c/(r-g)\).
• Example: the value of getting $100 next year and 8% more each year afterwards, when the rate of return is 10%, is

\[\frac{100}{0.10-0.08} = \frac{100}{0.02} = 5000\]